Atomic multipoles: Electrostatic potential fit,local reference axis systems,and conformational dependence

Currently, all standard force fields for biomolecular simulations use point charges to model intermolecular electrostatic interactions. This is a fast and simple approach but has deficiencies when the electrostatic potential (ESP) is compared to that from ab initio methods. Here, we show how atomic multipoles can be rigorously implemented into common biomolecular force fields. For this, a comprehensive set of local reference axis systems is introduced, which represents a universal solution for treating atom‐centered multipoles for all small organic molecules and proteins. Furthermore, we introduce a new method for fitting atomic multipole moments to the quantum mechanically derived ESP. This methods yields a 50–90% error reduction compared to both point charges fit to the ESP and multipoles directly calculated from the ab initio electron density. It is shown that it is necessary to directly fit the multipole moments of conformational ensembles to the ESP. Ignoring the conformational dependence or averaging over parameters from different conformations dramatically deteriorates the results obtained with atomic multipole moments, rendering multipoles worse than partial charges. © 2012 Wiley Periodicals, Inc.     

Partial charges by multipole constraint. Application to the amino acids

Atom-centered partial charges which exactly reproduce the lowest several multipoles of a molecule's charge distribution can be obtained in a straightforward and convenient manner from the output of existing electronic structure calculations. The multipole constraint method is demonstrated by a computation of partial charges for the twenty common amino acids. The electron density employed here, derived from a semiempirical MNDO calculation, incorporates Slater-type orbitals which imbue it with the exponential fall-off vital to electronic tunneling calculations. In addition, a procedure based on these charges is described which divides the original electron density into two components, a large component with a simple electrostatic potential, and a much smaller residual whose several lowest multipoles vanish.     

Rules for minimal atomic multipole expansion of molecular fields

A nonempirical minimal atomic multipole expansion (MAME) defines atomic charges or higher multipoles that reproduce electrostatic potential outside molecules. MAME eliminates problems associated with redundancy and with statistical sampling, and produces atomic multipoles in line with chemical intuition.     

Multipole correction of atomic monopole models of molecular charge distribution. I. Peptides

The defects in atomic monopole models of molecular charge distribution have been analyzed for several model-blocked peptides and compared with accurate quantum chemical values. The results indicate that the angular characteristics of the molecular electrostatic potential around functional groups capable of forming hydrogen bonds can be considerably distorted within various models relying upon isotropic atomic charges only. It is shown that these defects can be corrected by augmenting the atomic point charge models by cumulative atomic multipole moments (CAMMs). Alternatively, sets of off-center atomic point charges could be automatically derived from respective multipoles, providing approximately equivalent corrections. For the first time, correlated atomic multipoles have been calculated for N-acetyl, N'-methylamide-blocked derivatives of glycine, alanine, cysteine, threonine, leucine, lysine, and serine using the MP2 method. The role of the correlation effects in the peptide molecular charge distribution are discussed.     

The large quadrupole of water molecules

Many quantum mechanical calculations indicate water molecules in the gas and liquid phase have much larger quadrupole moments than any of the common site models of water for computer simulations. Here, comparisons of multipoles from quantum mechanical∕molecular mechanical (QM∕MM) calculations at the MP2∕aug-cc-pVQZ level on a B3LYP∕aug-cc-pVQZ level geometry of a waterlike cluster and from various site models show that the increased square planar quadrupole can be attributed to the p-orbital character perpendicular to the molecular plane of the highest occupied molecular orbital as well as a slight shift of negative charge toward the hydrogens. The common site models do not account for the p-orbital type electron density and fitting partial charges of TIP4P- or TIP5P-type models to the QM∕MM dipole and quadrupole give unreasonable higher moments. Furthermore, six partial charge sites are necessary to account reasonably for the large quadrupole, and polarizable site models will not remedy the problem unless they account for the p-orbital in the gas phase since the QM calculations show it is present there too. On the other hand, multipole models by definition can use the correct multipoles and the electrostatic potential from the QM∕MM multipoles is much closer than that from the site models to the potential from the QM∕MM electron density. Finally, Monte Carlo simulations show that increasing the quadrupole in the soft-sticky dipole-quadrupole-octupole multipole model gives radial distribution functions that are in good agreement with experiment.     

A multipole approximation of the electrostatic potential of molecules

The problem of approximating three-dimensional spatial distributions of quantum-mechanical electrostatic potentials of molecules by analytic potentials on the basis of atomic charges, real dipoles, and atomic multipoles up to quadrupoles inclusive was considered. Real dipole potentials are created by pairs of point charges of opposite signs, and the search for their arrangement in the volume of a molecule is part of the approximation problem. A FitMEP program was developed for the optimization of the parameters of models of the types specified taking into account molecular symmetry. It was shown for the example of several molecules (HF, CO, H₂O, NH₃, CH₄, formaldehyde, methanol, formamide, ethane, cyclopropane, cyclobutane, cyclohexane, tetrahedrane, cubane, adamantane, ethylene, and benzene) that the real dipole and atomic multipole models gave errors in approximated quantum-mechanical electrostatic potential values smaller by one or two orders of magnitude compared with the atomic charge model. The atomic charge model was shown to be virtually inoperative as applied to saturated hydrocarbons. Real dipole models were slightly inferior to atomic multipole models in quality but had all the advantages of the potential of point charges as concerned simplicity and compactness, and their use in potential energy calculations did not require changes in the existing program codes.     

A multipole-based water potential with implicit polarization for biomolecular simulations

A new water potential, DMIP (distributed multipoles, implicit polarization), is constructed using distributed multipoles to describe the electrostatic interactions, while accounting for polarization implicitly. In this procedure, small clusters are randomly sampled from atomistic simulations of bulk water using the AMOEBA (Ren and Ponder, J Comput Chem 2002, 23, 1497) potential. The multipole moments of the central water in each cluster are obtained from ab initio densities for each cluster, and the moments are then averaged over all clusters. Properties of bulk water calculated using DMIP compare favorably with existing data from AMOEBA simulations and experiment, with a conservative estimate of reduction in compute time of roughly 40%. The implicit force-field is also shown to work compatibly with existing polarizable multipole-based force-fields for biomolecules.     

Force related atomic multipoles in planar molecules. Derivation of atomic quadrupole and octupole moments

Atomic multipoles as defined by current methods generally do not account for forces in molecules that arise from external electrostatic fields. It is pointed out that such forces and the electrostatic potential that the molecule itself generates are both determined by the molecular multipolar tensors. The latter constitute therefore the fundamental molecular constants that determine the molecular electrostatics apart from polarization. In general the multipolar tensors include contributions from the atomic multipoles and their fluxes. In planar molecules, however, the perpendicular charge flux is zero by symmetry. This gives rise to a (previously introduced) formalism that extracts analytical, force-related, atomic multipoles from the molecular multipolar tensors. This formalism is extended in this work to include force-related (FR) atomic quadrupoles and octupoles in planar molecules. The properties of the FR atomic multipoles, including their perpendicular fluxes, are discussed and some formal theoretical and computational advantages that characterize them are indicated. As an example, the electrostatics of OCS, including the molecular electrostatic potential and the forces on the nuclei due to an external point charge, is discussed.     

Comparison of methods for point‐charge representation of electrostatic fields

The calculation of the electrostatic potential resulting from an infinite or extended array of charges in the interior of a region of interest is a frequent task in computational chemistry. In case of a periodic potential this can, for example, be done by Ewald summation or by multipole methods. An important alternative are those methods where arrays of auxiliary point charges are optimized with respect to charge and/or position to reproduce the original electrostatic potential. In the literature different variations are reported. We compare the performance of some of these with respect to their ability to reproduce the original potential and the computational effort required. Between (1) surface charges determined by the conductor‐boundary condition, (2) optimized surface charges, and (3) surface charges floating on the surface we find that (2) offers good quality with small computational costs involved. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Using multipole point charge distributions to provide the electrostatic potential in the variational explicit polarization (X-Pol) potential

The equations defining the variational explicit polarization (X-Pol) potential introduced in earlier work are modified in the present work so that multipole point charge distributions are used instead of Mulliken charges to polarize the monomers that comprise the system. In addition, when computing the electrostatic interaction between a monomer whose molecular orbitals are being optimized and a monomer whose electron density is being used to polarize the first monomer, the electron densities of both monomers are represented by atom-centered multipole point charge distributions. In the original formulation of the variational X-Pol potential, the continuous electron density of the monomer being optimized interacts with external Mulliken charges, but this corresponds to the monopole truncation in a multipole expansion scheme in the computation of the Fock matrix elements of the given monomer. The formulation of the variational X-Pol potential introduced in this work (which we are calling the “multipole variational X-Pol potential”) represents the electron density of the monomer whose wave function is being variationally optimized in the same way that it represents the electron densities of external monomers when computing the Coulomb interactions between them.     

Atomic forces for geometry‐dependent point multipole and Gaussian multipole models

In standard treatments of atomic multipole models, interaction energies, total molecular forces, and total molecular torques are given for multipolar interactions between rigid molecules. However, if the molecules are assumed to be flexible, two additional multipolar atomic forces arise because of (1) the transfer of torque between neighboring atoms and (2) the dependence of multipole moment on internal geometry (bond lengths, bond angles, etc.) for geometry‐dependent multipole models. In this study, atomic force expressions for geometry‐dependent multipoles are presented for use in simulations of flexible molecules. The atomic forces are derived by first proposing a new general expression for Wigner function derivatives . The force equations can be applied to electrostatic models based on atomic point multipoles or Gaussian multipole charge density. Hydrogen‐bonded dimers are used to test the intermolecular electrostatic energies and atomic forces calculated by geometry‐dependent multipoles fit to the ab initio electrostatic potential. The electrostatic energies and forces are compared with their reference ab initio values. It is shown that both static and geometry‐dependent multipole models are able to reproduce total molecular forces and torques with respect to ab initio, whereas geometry‐dependent multipoles are needed to reproduce ab initio atomic forces. The expressions for atomic force can be used in simulations of flexible molecules with atomic multipoles. In addition, the results presented in this work should lead to further development of next generation force fields composed of geometry‐dependent multipole models. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Base stacking in cytosine dimer. A comparison of correlated ab initio calculations with three empirical potential models and density functional theory calculations

Ab initio MP2/6-31G* interaction energies were calculated for more than 80 geometries of stacked cytosine dimer. Diffuse polarization functions were used to properly cover the dispersion energy. The results of ab initio calculations were compared with those obtained from three electrostatic empirical potential models, constructed as the sum of a Lennard-Jones potential (covering dispersion and repulsion contributions) and the electrostatic term. Point charges and point multipoles of the electrostatic term were also obtained at the MP2/6-31G* level of theory. The point charge MEP model (atomic charges derived from molecular electrostatic potential) satisfactorily reproduced the ab initio data. Addition of π-charges localized below and above the cytosine plane did not affect the calculated energies. The model employing the distributed multipole analysis gave worse agreement with the ab initio data than the MEP approach. The MP2 MEP charges were also derived using larger sets of atomic orbitals: cc-pVDZ, 6-311 + G(2d, p), and aug-cc-pVDZ. Differences between interaction energies calculated using these three sets of point charges and the MP2/6-31G* charges were smaller than 0.8 kcal/mol. The correlated ab initio calculations were also compared with the density functional theory (DFT) method. DFT calculations well reproduced the electrostatic part of interaction energy. They also covered some nonelectrostatic short-range effects which were not reproduced by the empirical potentials. The DFT method does not include the dispersion energy. This energy, approximated by an empirical term, was therefore added to the DFT interaction energy. The resulting interaction energy exhibited an artifact secondary minimum for a 3.9-4.0 vertical separation of bases. This defect is inherent in the DFT functionals, because it is not observed for the Hartree-Fock + dispersion interaction energy.© 1996 John Wiley & Sons, Inc.     

Gaussian induced dipole polarization model

A new induced dipole polarization model based on interacting Gaussian charge densities is presented. In contrast to the original induced point dipole model, the Gaussian polarization model is capable of finite interactions at short distances. Aspects of convergence related to the Gaussian model will be explored. The Gaussian polarization model is compared with the damped Thole-induced dipole model and the point dipole model. It will be shown that the Gaussian polarization model performs slightly better than the Thole model in terms of fitting to molecular polarizability tensors. An advantage of the model based on Gaussian charge distribution is that it can be easily generalized to other multipole moments and provide effective damping for both permanent electrostatic and polarization models. Finally, a method of parameterizing polarizabilities is presented. This method is based on probing a molecule with point charges and fitting polarizabilities to electrostatic potential. In contrast to the generic atom type polarizabilities fit to molecular polarizability tensors, probed polarizabilities are significantly more accurate in terms of reproducing molecular polarizability tensors and electrostatic potential, while retaining conformational transferability.     

Charges derived from distributed multipole series

A novel method to calculate electrostatic charges is proposed. Partial charges are fitted to reproduce the electrostatic potential of a distributed multipole series without explicitly evaluating electrostatic potentials. The calculation is economical and results in charges reflecting the symmetry properties of charge centers and being independent of molecular orientation.     

Topological analysis of the molecular charge density and impact sensitivy models of energetic molecules

Important explosives of practical use are composed of nitroaromatic molecules. In this work, we optimized geometries and calculated the electron density of 17 nitroaromatic molecules using the Density Functional Theory (DFT) method. From the DFT one-electron density matrix, we computed the molecular charge densities, thus the electron densities, which were then decomposed into electric multipoles located at the atomic sites of the molecules using the distributed multipole analysis (DMA). The multipoles, which have a direct chemical interpretation, were then used to analyze in details the ground state charge structure of the molecules and to seek for correlations between charge properties and sensitivity of the corresponding energetic material. The DMA multipole moments do not present large variations when the size of the Gaussian basis set is changed; the largest variations occurred in the range 10-15% for the dipole and quadrupole moments of oxygen atoms. The charges on the carbon atoms of the aromatic ring of each molecule become more positive when the number of nitro groups increases and saturate when there are five and six nitro groups. The magnitude and the direction of the dipole moments of the carbon atoms, indicators of site polarization, also depend on the nature of adjacent groups, with the largest dipole value being for C-H bonds. The total magnitude of the quadrupole moment of the aromatic ring carbon atoms indicates a decrease in the delocalized electron density due to an electron-withdrawing effect. Three models for sensitivity of the materials based on the DMA multipoles were proposed. Explosives with large delocalized electron densities in the aromatic ring of the component molecule, expressed by large quadrupole values on the ring carbon atoms, correspond to more insensitive materials. Furthermore, the charges on the nitro groups also influence the impact sensitivity.     

On combining Thole's induced point dipole model with fixed charge distributions in molecular mechanics force fields

The Thole induced point dipole model is combined with three different point charge fitting methods, Merz–Kollman (MK), charges from electrostatic potentials using a grid (CHELPG), and restrained electrostatic potential (RESP), and two multipole algorithms, distributed multipole analysis (DMA) and Gaussian multipole model (GMM), which can be used to describe the electrostatic potential (ESP) around molecules in molecular mechanics force fields. This is done to study how the different methods perform when intramolecular polarizability contributions are self‐consistently removed from the fitting done in the force field parametrization. It is demonstrated that the polarizable versions of the partial charge models provide a good compromise between accuracy and computational efficiency in describing the ESP of small organic molecules undergoing conformational changes. For the point charge models, the inclusion of polarizability reduced the the average root mean square error of ESP over the test set by 4–10%. © 2015 Wiley Periodicals, Inc.